This competition is part of the course 10316 Materials design with machine learning and artificial intelligence at DTU January 2022. During the first two weeks you have learned the basics of programming and machine learning, and now it's time to apply these skills in a real world problem.
\nThe objective of the competition is to build a machine learning model that predicts the elastic tensor components of 2D materials.
\nFor elastic materials, the linear elasticity can be described by the general equation for Hooke’s law:
\n\\[ \\boldsymbol{\\sigma} = \\boldsymbol{C} : \\boldsymbol{\\varepsilon} \\]
\nor in engineering notation:
\n\\[ \\sigma_{ij} = C_{ijkl} \\varepsilon_{kl}\\]
\nwhere \\( \\boldsymbol{\\sigma} \\) is the stress tensor, \\( \\boldsymbol{\\varepsilon} \\) is the strain tensor and \\( \\boldsymbol{C} \\) is the fourth-order stiffness tensor. For 2D materials there are 3 unique relations between stress and strain:
\n\\[\\begin{align} \\begin{bmatrix} \\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{12} \\end{bmatrix} &= \\begin{bmatrix} C_{1111} & C_{1122} & C_{1112} \\\\ C_{2211} & C_{2222} & C_{2212} \\\\ C_{1211} & C_{1222} & C_{1212} \\end{bmatrix} \\cdot \\begin{bmatrix} \\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{12} \\end{bmatrix} \\\\ &= \\begin{bmatrix} C_{1111} & C_{1122} & C_{1112} \\\\ & C_{2222} & C_{2212} \\\\ sym & & C_{1212} \\end{bmatrix} \\cdot \\begin{bmatrix} \\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{12} \\end{bmatrix} \\\\ &= \\begin{bmatrix} C_{11} & C_{12} & C_{13} \\\\ & C_{22} & C_{23} \\\\ sym & & C_{33} \\end{bmatrix} \\cdot \\begin{bmatrix} \\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{12} \\end{bmatrix} \\end{align}\\]
\nThe bulk modulus (Voigt definition) can be calculated as
\n\\[ K_V = \\frac{1}{4} (C_{11}+C_{22}+2C_{12})\\]
\nwhich is the prediction objective for this competition.
\nThe dataset for the competition consists of 3858 2D materials along with their DFT-calculated elastic components. The structures are provided as Atoms objects containing the atomic numbers and positions for the structure. The dataset is divided into two sets (85% for training and 15% for testing). For the test set, only the structures and energy are provided (and not the elastic components).
", "isPublished": false, "hasIsPublished": false, "postTitle": "", "hasPostTitle": false, "mimeType": "", "hasMimeType": false }, { "name": "Evaluation", "content": "The evaluation metric for this competition is RMSE-Score (Root-mean-squared-error). The RMSE score is a commonly used measure of the differences between of observed values and predicted values by an estimator or model. The RMSE score is given by:
\n\\[ RMSE = \\sqrt{\\frac{\\sum_i (\\hat{y}_i-y_i)^2}{N}} \\]
\nwhere i runs over samples in the test set and N is the number of samples.
\nFor every structure in the dataset the submission file should contain a row with 2 columns. The first column is the structure ID and the second column is the predicted elastic property.
\nID,Voigt_modulus\n
1,0
2,0
3,0